Originally a standard for HD (high-definition) television, 1080p now generally refers to an image size of 1920 pixels wide by 1080 pixels high.
In television and video, 1080p specifies not only 1920 x 1080 pixels, but progressive encoding, which means that each frame of video include all 1080 "lines" of the picture. The alternative is 1080i, where the "i" stands for interlaced, meaning that each frame includes only every other line.
Interlaced video makes little sense for digital and mobile settings, so most phones record progressive (1080p or 720p) video instead.
1080p is also sometimes used to refer to display resolution of 1920 x 1080 pixels. In that context, the "p" doesn't mean anything, but is still used for consistency with televisions of the same resolution.
In still image terms, 1080p is 2 megapixels.
1080p is higher-resolution (and therefore delivers more detail) than 720p.
See: 720p
The next step above 1080p is 4K.
See: 4K